The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

In nonlinear sciences, the research of acoustic waves in shallow water is a widespread topic 33 .This type of acoustic wave is seen in streams, ocean coastlines, desert, and the water and can be explained by the Boussinesq model 34 , which is detailed below: where x, y, z and t are spatial and temporal terms, Q is the dependent variable of the governing model.The Boussinesq model was initially established by Joseph Boussinesq in 1872 to explain how long, small-amplitude waves flow at a constant pace in a water channel with a constant depth.Additionally; the Boussinesq model is extensively applied in marine and coastal engineerings 35 .Because the Boussinesq model accurately models viscous flows involving various fluids with interfaces between them, the authors have considered it.Similar to viscous stresses, this model offers a fair distribution of turbulent stresses related to mean velocity gradients.It is also applied in road and geotechnical engineering for problems involving the distribution of vertical tension in soil medium.Because Boussinesq's approximation ignores density fluctuations in inertial terms, it is frequently used in fluid dynamics with mild density gradients, which makes it appropriate for some applications.
The Boussinesq model is frequently used in coastal and ocean engineering to predict wave propagation in shallow coastlines and seas 36 .Now comprehend the dynamics of wave propagations on the ocean surfaces, our goal is to construct the exact solutions to a novel integrable (3+1)-dimensional Boussinesq model by implementing the unified Riccati equation expansion technique 37 .We take into consideration the innovative (3+1)-dimensional Boussinesq model 38 to accomplish these goals.
where α, β, γ and σ are constants and (x, y, z) and t shows the spatial and temporal parameters, respectively.By employing Hirota bilinear technique in 2019, Wazwaz and Kaur 39 created several solution to Eq. ( 2) and then used the exp(−�(ζ ))-expansion technique to establish certain analytical solution.It is crucial to note that Eq. ( 2) can be transformed to the typical fourth-order Boussinesq model, which is provided by Eq. (1), if the value of σ = 0 in Eq. ( 2), then transforms into a new integrable (2+1)-dimensional Boussinesq model.Thus, if we take into account β = 1, α = 1, γ = 1 and σ = 1 , then Eq. ( 2), become In this work, Eq. ( 3) can be transformed to (2+1)-dimensional Boussinesq model when z = x and z = y .The lump solution to various dimensionally reduced equations have previously been discovered by some researchers [40][41][42] .In particular, Kaur and Wazwaz 28 explored the lump solution to the reduced equations of (3+1)-dimensional generalized Boussinesq model.Singh et al. 41 found results for the novel (2+1)-dimensional Boussinesq model, including both brilliant and dark rogue waves.Huang et al. 42 discovered the lump waveform and kink soliton solution for the generalized (3+1)-dimensional KP model.The physical phenomena can be enhanced by using the visual representation of the solutions to the Boussinesq model, which highlights the physical characteristics of the model.Understanding the solutions to real-world problems, such as ocean waves, aids us in accurately comprehending them.It is important to emphasize that all results show novel properties emerging in shallow water waves.
Our literature analysis indicates that the UREE technique has not been previously employed to investigate the transmission of optical solitons in optical fibers.In our analysis, we have also considered bifurcation analysis, physical features, and the application of findings to better understand the dynamic processes of the model.Through the extension of the Riccati equation, a methodical technique is employed in the process.An approximate version of the solutions known as the general ansatz is used that subsequently entered into the formula.The differential equation is converted into a nonlinear algebraic equation using algebraic operations.After that, new soliton solutions can be obtained by solving these nonlinear algebraic equations.Our objective for this study is to evaluate the soliton solutions of the (3+1)-dimensional Boussinesq model using a novel technique.These are the primary sections of this research.The unified Riccati equation expansion (UREE) method is described in Section "Description of the UREE technique".The derivation of the soliton solutions has been stated in Section "Implementation".In Section "Bifurcation analysis", the phase portraits of bifurcations are utilized to illustrate the qualitative behavior of the analyzed model.In Section "Results and discussions", we present some discussions on the obtained results.Finally, Section "Conclusion" covers the conclusion part of this study.

Description of the UREE technique
The generic form of NLEEs is Step-i: Utilize the transformations of wave, such that be used in Eq. ( 4), then we can obtain the nonlinear ordinary differential equation (NLODE) as, (1)

Implementation
The main focus of this part is the application of our suggested technique to verify its performance, efficacy, and dependability.It will offer us a selection of solutions for the Boussinesq model, which is integrable in the (3+1)-dimensional space.The following transformation is provided in Eq. ( 5).Next, the transformation provided in Eq. ( 5) is applied to transform Eq. (3) into NLODE.Thus Integrating twice Eq. ( 14), to get the required second-order ordinary differential equation By using the balancing principle from Eq. ( 15), we can get M = 2 .Let M = 2 , the general solution of Eq. ( 7) becomes Now, equating the coefficients of same power of (φ(ξ )) d , where d = 0, 1, 2, 3, . . . .The Eq. ( 16) is incorporated into Eq.( 15) with Eq. ( 8) and thus we can obtain a system of algebraic equations such as www.nature.com/scientificreports/Using symbolic computing, we solve the above system of equations and hence we can obtain a set of solutions as follows Family-1: According to the results, Family-1 is satisfied by the following solutions.
• When L > 0 , we get

Family-2:
According to the results, Family-2 is satisfied by the following solutions.

Family-3:
According to the results, Family-3 is satisfied by the following solutions.
• When L > 0 , we get • When L < 0 , we get • When L = 0 , we get

Bifurcation analysis
In this section, we investigate Eq. (3) using bifurcation theory and phase portrait analysis 35 .The system described by Eq. ( 3) has its own dynamic influence, allowing us to see how changes in parameters affect the quality of the system.A framework for examining the bifurcations that occur within a family of systems is provided by bifurcation theory, which enables us to pinpoint the typical bifurcation patterns.By using the Galilean transformation, we may express the planar dynamical system for Eq. ( 15) as follows, When applying the aforementioned transformation to Eq. ( 15), we obtain www.nature.com/scientificreports/Assuming that q and p are the functions: As p ′ = Cq − Dq 2 , where C = 1 4 (−3 − 4m + 4m 2 + 4n) and D = 1.So that q ′ = p and q ′′ = Cq − Dq 2 .That is There are two equilibrium points for the equation p ′ = Cq − Dq 2 : J 1 (Q, P) = (0, 0) and J 2 (Q, P) = (Q, 0) .Therefore, (Q, 0) can be classified as a saddle point when J (Q, P) < 0 , a center when J (Q, P) > 0 and a cuspidal point when J (Q, P) = 0 .Similar to this, if J (Q, P) < 0 , J (Q, P) > 0 and J (Q, P) = 0 , then (0, p) is a saddle point, a center and a cuspidal point respectively.It is significant to remember that Q and P can take real values depending on the specific choices made for the parameters.We experience many circumstances for various parameter choices, each of which will be discussed in detail.
Case-4: For C > 0 and D < 0 , If C > 0 and D < 0 , then for only one equilibrium point exist that is J 1 (0, 0) .A = J 1 (Q, P) is a center point if A = J (Q, P) > 0 as shown in Fig. 9. Now add the perturbation term in Eq. ( 37), to discuss the chaotic, quasi-periodic and sensitivity analysis structures by using different values of frequency with initial conditions.Now, the chaotic equation below where is frequency, Ŵ is amplitude and ξ is independent variable.The illustration of quasi-periodic, chaotic and sensitivity analysis as shown in Fig. 10 under suitable parametric values.

Results and discussions
This section discuss on recent findings in conjunction with a comparison of previous study 38 where the authors investigated the integrable (3+1)-dimensional Boussinesq model to obtain lump solutions.In our work, we proposed the unified Riccati equation expansion method for (3+1)-dimensional Boussinesq model and minimized their dimensional models in shallow water waves to find dark, unique, periodic, and rational solutions.In nonlinear optics, the study of interaction between strong light beams and nonlinear materials is relevant to dark, bright solitons.They are pertinent to phenomena like pulse compression and the creation of supercontinuous.To solve distinct nonlinear partial differential equations that arise in mathematical physics, soliton solutions including dark and brilliant solitons are employed.These solutions shed light on the behaviour of complicated systems.Furthermore, bifurcation analysis exposes every possible nonlinear dynamic system phase portrait.Exact soliton solution structures are shown in Figs. 1, 2, 3, 4 and 5, and phase pictures from bifurcation analysis are displayed in Figs. 6, 7, 8 and 9. Figure 10 represents the chaotic structure with quasi-periodic and sensitivity analysis.The dark soliton solutions are quieter solitary waves than the background whereas the singular soliton 1 4 (−3 − 4m + 4m 2 + 4n)q − q 2 − p ′ = 0, (37) (38) q ′ = H(q, p), p ′ = G(q, p).

Conclusion
In the present work, we examined the soliton solutions of shallow water waves in a (3+1)-dimensional Boussinesq model.We conducted a qualitative study of the proposed model and derived accurate solutions using the UREE method.We successfully derived dark, solitary, periodic, and rational solutions.We study these solutions with phase depictions to attain a better understanding of the theory of motivation.We utilize bifurcation and chaos theories to comprehend the planar dynamical system and showcase its dependence on physical parameters like quasi-periodic and sensitivity analysis.These innovative results encourage new understandings of wave motion dynamics in mathematical simulations.Using symbolic computing, we solve nonlinear wave problems in multiple fields, such as mathematical physics and engineering.These findings could contribute to understanding how waves propagate in shallow water in oceanography.Future studies on the particular (3+1)-dimensional Boussinesq model may focus on breather wave solutions, hybrid formulations, rogue waves, and multi-lump waveforms.
Ŵξ ), www.nature.com/scientificreports/solutionsarediscontinuousderivativessuch as peakons and compactions.Periodic solutions repeat throughout time and help with pattern recognition and understanding equation structure.Figures6, 7, 8 and 10 phase pictures are created by choosing appropriate values from Eq.(37).Solutions formed from equilibrium points and parameter values are shown by variable A, where A > 0 shows the center point, A < 0 is the saddle point, and A = 0 is the cuspidal point.